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PSI AP Physics C – Work and Energy (With Calculus) Multiple Choice Questions 1. An object moves according to the function x = t7/2 where x is the distance traveled and t is the time. Its kinetic energy is proportional to: (A) t5/2 (B) t7/2 (C) t5 (D) t3/2 (E) t7 2. Which of the following best describes the relationship between force and potential energy? (A) Force is the anti-derivative of potential energy. (B) Force is the negative gradient of potential energy. (C) Potential energy is the negative gradient of force. (D) Potential energy is the derivative of force. (E) Force is the anti-derivative of potential energy. 3. A 5-kilogram ball moves in the x direction where x represents the ball’s position. The potential energy U of the ball in Joules is given as a function: U(x) = 4x2 - 3x + 2. The force on the particle at x = 4 m is: (A) 29 N in -x direction (B) 29N in +x direction (C) 108 N in -x direction (D) 45 N in -x direction (E) 108 N in +x direction 4. A student pushes a box across a rough, flat surface at a constant speed v. The box has a mass m, and the coefficient of sliding friction is represented by µ. The power supplied by the person to the box is (A) 0 (B) µmg/v (C) µv/mg (D) mg/µv (E) µmgv 5. The force exerted by a spring is given by: F = 𝑘𝑥 4 2 . If k is 100 N/m, find the work done by the spring on a mass from x = 0 m to x = 2 m. (A) 100 J (B) 320 J (C) 800 J (D) 1600 J (E) 2400 J 6. A man lifts a mass m at constant speed to a height h in time t. How much work is done by the weight lifter on the mass? (A) mgt (B) zero (C) mgh (D) mgh/t (E) mgt/h 7. A spring force is given by the formula F = 20x - 12x2, where F is in N and x is in m. What is the change in potential energy when the spring is stretched 3 m from its equilibrium position? (A) 18 J (B) -18 J (C) 56 J (D) -56 J (E) 64 J 8. On top of a skyscraper of height H, a ball of mass m is thrown directly downward with an initial speed vo. What is the speed of the ball before it strikes the ground? Ignore air resistance. 1 1 (A) mgh- 𝑚(vo2-vf2) (B) mgh+ 𝑚(vo2+vf2) 2 (D) 2 √𝑣𝑜2 + 2𝑔ℎ 1 (E) 2 𝑚(vf2 1 (C) mgh+ 𝑚(vf2 -vo2) 2 -vo2)-mgh 9. A ball attached to a string rotates in a complete circle at a constant speed. The work done during each revolution is: (A) 0 (B) U (C) U+Ke (D) Ke (E) Ke-U 10. The potential energy of two molecules is given by: U= 2 𝑟7 4 − 5. If r is the distance 𝑟 between two molecules what is the force acting on the particles if r=1 m? (A) 0.75 N (B) 0.67 N (C) 2 N (D) -6 N (E) 10 N 11. A force of 40 N compresses a spring with a spring constant 80 N/m. How much energy is stored in the spring? (A) 10 J (B) 15 J (C) 20 J (D) 25 J (E) 30 J 12. What is the power delivered by gravity to a 6 kg block 4 s after it has fallen from rest? (A) 2400 W (B) 1000 W (C) 800 W (D) 1200 W (E) 2000 W 13. If F(x) = 8x3-3x2 what is the work done from x =1 m to x =2 m? (A) 0.5 J (B) 0.8 J (C) 2 J (D) 12 J (E) 23 J 14. A 2 kg block is pushed horizontally across a rough surface with a coefficient of kinetic friction of 0.2, at a constant speed of 4 m/s, by a force F. The work that is done by the force in 5 s is: (A) 20 J (B) 40 J (C) 60 J (D) 80 J (E) 100 J 15. A constant force supplies an average power of 8 W to a box during a certain time interval. If the box has an average speed of 4 m/s, and the force acts in the same direction as motion of the object, the magnitude of the force is: (A) 1 N (B) 2 N (C) 4 N (D) 6 N (E) 8 N 16. Which of the following is true about an oscillating system in SHM? (A) Potential energy is never equal to kinetic energy. (B) Potential energy is equal to kinetic energy at all points. (C) Potential energy decreases all the time. (D) Kinetic energy increases all the time. (E) Maximum potential energy is equal to maximum kinetic energy. 1 17. A 4 kg mass is moving with a velocity given by v (t) = 4t4. At t= 1 s, the instantaneous power delivered by the net force is: (A) 1 W (B) 3 W (C) 12 W (D) 14 W (E) 20 W 18. What is the instantaneous power delivered by the net force at t = 2 s to a 2 kg mass 1 moving according to x(t) = 3t3? (A) 2 W (B) 10 W (C) 16 W (D) 24 W (E) 32 W 19. A particle of mass m follows the potential energy graph as shown above. The particle is initially at rest at RO. Determine the particle’s speed at position 4Ro (A) U0 m (B) 2U 0 m (C) 4U 0 m (D) 6U 0 m (E) 8U 0 m 20. The function U(r) = ar-3/2 + b represents the potential energy of a particle, where a and b are positive constants, which of the following is an expression for the force on the particle? (A) 5 3 𝑎𝑟 −2 2 (B) 2𝑎𝑟 − 1 2 (C)− 3 2 5 1 −5 (D)− 2 𝑎𝑟 2 𝑎𝑟 −2 (E) 1 2 3 𝑎𝑟 −2 21. A mass moves under the influence of a potential energy given by: U(x) = x3 – 2x2 At x = 2 m, the force on the mass will be: (A) 4 N, + direction (B) 4 N, - direction (C)3 N, + direction (D) 3 N, - direction (E) 7 N, + direction 22. A 6 kg object’s potential energy is represented by: U(r) = 9r2+4. Find the acceleration of the object at r = 2 m. (A) -4 m/s2 (B) 4 m/s2 (C) -6 m/s2 (D) 6 m/s2 (E) 12 m/s2 23. A restoring force F = -2x3 acts on an object, where x is the displacement of the object from its equilibrium position. How much work must be done by an external force to move the object from x= 0 to x= 2 m? A) 2 J B) 24 J C) 8 J D) 10 J E) 15 J 24. How much work is done by an external force in stretching a spring from x = 0 to x = 0.2 m, whose restoring force varies according to the following formula: F(x) = -(6 N/m)x + (30 N/m2)x2? (A) -0.04 J (B) 0.04 J (C) -0.16 J (D) 0.16 J (E) 0.08 J 25. The restoring force of a spring varies with displacement according to: F = -kx2. What is the potential energy of the spring when it is stretched by a distance D from equilibrium? (A) 𝑘𝐷 2 3 (B) 𝑘𝐷 2 4 (C) 𝑘𝐷 3 3 (D) 𝑘𝐷 2 2 (E) 𝑘𝐷 3 2 1 26. The potential energy of a spring is given by the following formula: U(x) = 3αx3 – βx, where α and β are positive constants. Which of the following represents the restoring force of the spring? (A) 3αx2 – β (B) -3αx2 + β (C) 2αx3 - βx (D) 3αx - β (E) -αx2 + β 1 1 27. The potential energy of a spring is given by the following formula: U(x) = 3αx3 – 2βx2, where α and β are positive constants. What is the location(s) of the equilibrium points? (A) x=0 only 𝛽 (B) x= 𝛼 only 𝛽 (C) x=0 and x=𝛼 only 2𝛽 (D) x=0 and x= 𝛼 only 3𝛽 (E) x=2𝛼 only 28. The gravitational force between a spaceship and Earth is given by the formula: F=− 𝐺𝑀𝑚 𝑟2 . Which of the following represents the potential energy of the spaceship/Earth system, assuming that U = 0 when r→ ∞. (A) U = (B) U = (C) U = (D) U = 𝐺𝑀𝑚 𝑟3 𝐺𝑀𝑚 𝑟2 𝐺𝑀 𝑟 𝐺𝑀𝑚 𝑟 𝐺𝑀𝑚 (E) U = − 𝑟 29. A conservative force parallel to the x-axis moves a particle along the x-axis. The 1 potential energy of the particle is given by U(x) = 4 𝛼𝑥 4 , where α = 0.5 J/m4. What is the force on the particle when it is located at x = 2 m? (A) -4 N (B) 4 N (C) 8 N (D) -8 N (E) 10 N 30. A conservative force parallel to the x-axis moves a particle along the x-axis. The 𝛼 potential energy of the particle is given by U(x) =− 𝑥 2 , where α is a positive constant. Which of the following represents the force on the particle? (B) 2𝛼 𝑥 (B) − 2𝛼 𝑥 (C) 3𝛼 𝑥 (D) 2𝛼 𝑥2 (E) − 2𝛼 𝑥3 31. A conservative force parallel to the x-axis moves a particle along the x-axis. The potential energy as a function of position is presented by the graph. The particle is released from rest at point A. What is the approximate force on the particle when it passes point B? (A) 3 N (B) 1 N (C) -3 N (D) 6 N (E) -6 N 32. A conservative force parallel to the x-axis moves a particle along the x-axis. The potential energy as a function of position is presented by the graph. The particle is released from rest at point A. At what location(s) is the particle is in stable equilibrium? (A) x = 1.0 m only (B) x = 2.0 m only (C) x = 4.0 m only (D) x = 2.0 m and x = 4 m only (E) x = 2.0 m and x = 5.6 m only 33. A conservative force parallel to the x-axis moves a particle along the x-axis. The potential energy as a function of position is presented by the graph. The particle is released from rest at point A. What is the largest value of x reached by the particle during this motion? (A) 5.6 m (B) 4.0 m (C) 6.9 m (D) 8.0 m (E) 2.0 m 34. A conservative force parallel to the x-axis moves a particle along the x-axis. The potential energy as a function of position is presented by the graph. The particle is released from rest at point A. At what location(s) is the particle is in unstable equilibrium? (A) x = 1.0 m only (B) x = 2.0 m only (C) x = 4.0 m only (D) x = 2.0 m and x = 4 m only (E) x = 2.0 m and x = 5.6 m only 35. A conservative force parallel to the x-axis moves a particle along the x-axis. The potential energy as a function of position is presented by the graph. The particle is released from rest at point A. What is the approximate force on the particle when it passes point D? (A) 2 N (B) -2 N (C) -3 N (D) -4 N (E) -6 N Answer Key 1.C 6.C 11.A 16.E 21.B 26.E 31.A 2.B 7.A 12.D 17.A 22.C 27.C 32.E 3.A 8.D 13.E 18.E 23.C 28.E 33.C 4.E 9.A 14.D 19.D 24.B 29.A 34.C 5.B 10.D 15.B 20.A 25.C 30.E 35.B